The optimization process and recommendations:

1. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
2. Explicitly ORDER BY After GROUP BY (modified query below): By default, the database sorts all 'GROUP BY col1, col2, ...' queries as if you specified 'ORDER BY col1, col2, ...' in the query as well. If a query includes a GROUP BY clause but you want to avoid the overhead of sorting the result, you can suppress sorting by specifying 'ORDER BY NULL'.
3. Prefer Sorting/Grouping By The First Table In Join Order (modified query below): The database can use indexes more efficiently when sorting and grouping using columns from the first table in the join order. The first table is determined based on the prediction of the the optimal first table, and is not necessarily the first table shown in the FROM clause.
4. Replace Join With Exists To Avoid Redundant Grouping (modified query below): When a joined table isn’t used anywhere other than in the WHERE clause, it's equivalent to an EXISTS subquery, which often performs better. In cases where the DISTINCT or GROUP BY clause contains only columns from the Primary key, they can be removed to further improve performance, as after this transformation, they are redundant.

Optimal indexes for this query:

ALTER TABLE `account` ADD INDEX `account_idx_account_number` (`account_number`);
ALTER TABLE `account_transaction` ADD INDEX `account_transactio_idx_transaction_id` (`transaction_code_id`);
ALTER TABLE `transaction_code` ADD INDEX `transaction_code_idx_id` (`id`);

The optimized query:

SELECT
        t.id 
    FROM
        account_transaction t 
    WHERE
        (
            EXISTS (
                SELECT
                    1 
                FROM
                    transaction_code tc 
                JOIN
                    account a 
                WHERE
                    (
                        t.transaction_code_id = tc.id
                    ) 
                    AND (
                        t.account_number = a.account_number
                    )
            )
        ) 
    GROUP BY
        t.transaction_code_id 
    ORDER BY
        NULL